 This video is going to talk about simplifying fractions. Fractions can be written A over B, some number divided by another number, as long as B, the bottom number, is not equal to zero. So here are the different kinds of fractions that we can have. We can have proper fractions, and in a proper fraction the numerator, or the top part of the fraction, is going to be less than the denominator, or the bottom of the fraction. So that would be something like one-half, one is less than two. Improper fractions are going to be where we have just the opposite. The numerator number is going to be bigger, or greater than the denominator number. Oops, I spelled that wrong. And we'll just, you know it's this word here, the denominator. So it would be something like eleven over five. The top number is bigger than the bottom number. Mixed fractions have a whole number, and a fraction, plus a fraction. You've seen these things, like when you look at recipes it'll say two-and-one-half cups of flour, two-and-one-half would be a mixed fraction. A nice thing to know how to do is to be able to go back and forth between an improper fraction and a mixed fraction, because a lot of times in your homework you're going to have a mixed fraction that you have to put into an improper fraction, and sometimes there may be a case where you want to write an improper fraction to a mixed fraction. So I'm going to show you how to do both. We'll start with the improper fraction. I think it'll help us understand mixed, mixed to improper first. So if we go from improper to mixed, we can, we want to know how many times three goes into twenty-six perfectly. So we would take twenty-six and divide it by three, and it so happens that eight times three is twenty-four, so it goes in eight times perfectly, but there's two little pieces left over. So this mixed fraction would be eight whole parts, with two, and we have to write it over that same denominator, because that's the kind of fraction it is, it's the third fraction, so we have eight and two-thirds. Now with an improper fraction, there's a little trick that helps you be able to do it. Remember we were divided twenty-six by three and got eight going into it perfectly. Well if we go back, we can multiply eight times three, and then we would add the two more, and that would tell us how many thirds we have. So we have twenty-four plus two more, or just like we started with above, we have twenty-six-thirds. So a simplified fraction, what is a simplified fraction? Simplified fraction means that the numbers are going to have nothing in common. Okay, a simplified fraction, the numerator, I'm just going to use shorthand here, and denominator have no factors in common, no common factors. So good thing to do here would be just prime factor each of these numbers. So if I take twelve, I can say that that's two times six, and then that would be two times three, or you can look at this as four times three, there's another way of looking at twelve. And then fifteen would be three times five. Now I have everything prime factored, and then all I need to do is find the common prime factors top and bottom. And they both have a three, the top and the bottom have a three. So they are really a factor of one, and one times anything is just what you started with. We have two times two over five, or twelve-fifteen, so it's going to be equal to four-fifths. If we take 144, I'm going to prime factor over here because that's a big one, that would be two times seventy-two, and seventy-two would be two times thirty-six, and thirty-six would be two times eighteen, and eighteen would be two times nine, and nine would be three times three. That's a lot of factors. So two times two times two times two, one, two, three, four, and then two-threes. And then on the bottom, if we take fifty-four over here, that would be two times twenty-seven, and twenty-seven would be three times nine, three is the prime factor, and nine would be three times three. So we have two times three times three times three. So we've done the prime factors. Now we're looking for common factors top and bottom. So those cancel each other out, and this three will cancel out that three, a factor of one so that we can factor it out, and that three will cancel out that three, or factor out that three, and we have two times two times two over three, or eight over three. So 144 over fifty-four is eight over three. One last example that you'll have common in your homework, and this one is actually going to make me need my calculator. So when I do my calculator, the first thing that you can see is that if I take these two numbers, they both have these three zeros at the back. That's kind of like saying if I divide this thing by a hundred thousand, and divide this one by a hundred thousand, then it's really the same thing as just saying, okay, well I can cancel since they're all the same zeros. You normally can't do this with numbers, so I really am hesitant to say this, but if you have zeros, only zeros, then I can cancel out my zeros, and I really, or if I divide by a hundred thousand, if you're afraid you're going to make a mistake, then divide by the hundred thousand because they were both in the hundred thousands. I just have one ten divided by two eighty-six. One ten isn't too bad to prime factor. It's even, so we divide by two, and that gives us fifty-five, and we can see that that one's going to be divisible by five, and that happens to be five times eleven, and those are all prime numbers. We're going to say that that is two times five times eleven. When I look at two eighty-six, it's a little bit harder to see. It's kind of a big number, and actually there ends up being a number where I really don't know what it is. Well, I'll sort it out. Two eighty-six is even, so I'll divide by two, and then that gives me one forty-three, and now I'm stuck. It's not an even number. It doesn't divide by five, and I might divide by three because you see that three, but it really doesn't divide by three, so here's the trick. You take your number, one forty-three, and start dividing it by prime numbers. I know three doesn't work, two doesn't work, five doesn't work, seven is not going to work because if I do seven I'll go into fourteen, but then I'll have this three left over. So nine might work, but nine doesn't work because it gives me a decimal. So try again, one forty-three divided by the next prime would be eleven. And if I divide that by eleven, I get a perfect number of thirteen. So that means that one forty-three is eleven times thirteen, and now two eleven and thirteen are all prime numbers. So I have two times eleven times thirteen, and you can see that they both have eleven. So there's a factor of one, leaves me with two times five over two times thirteen, and two times five is ten, and two times thirte-oh, I have a common factor. Always double-check that. I had another common factor, so it's really going to be five over thirteen.